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Theorem xrmaxleim 11265
Description: Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
Assertion
Ref Expression
xrmaxleim  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )

Proof of Theorem xrmaxleim
Dummy variables  f  g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrlttri3 9810 . . . 4  |-  ( ( f  e.  RR*  /\  g  e.  RR* )  ->  (
f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
21adantl 277 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  (
f  e.  RR*  /\  g  e.  RR* ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
3 simplr 528 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  B  e.  RR* )
4 prid2g 3709 . . . 4  |-  ( B  e.  RR*  ->  B  e. 
{ A ,  B } )
53, 4syl 14 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  B  e.  { A ,  B }
)
6 simpllr 534 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  <_  B )
7 xrlenlt 8035 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
87ad3antrrr 492 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( A  <_  B  <->  -.  B  <  A ) )
96, 8mpbid 147 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  A )
10 breq2 4019 . . . . . . 7  |-  ( y  =  A  ->  ( B  <  y  <->  B  <  A ) )
1110notbid 668 . . . . . 6  |-  ( y  =  A  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1211adantl 277 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
139, 12mpbird 167 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  y )
14 xrltnr 9792 . . . . . 6  |-  ( B  e.  RR*  ->  -.  B  <  B )
1514ad4antlr 495 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  B )
16 breq2 4019 . . . . . . 7  |-  ( y  =  B  ->  ( B  <  y  <->  B  <  B ) )
1716notbid 668 . . . . . 6  |-  ( y  =  B  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1817adantl 277 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1915, 18mpbird 167 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  y )
20 elpri 3627 . . . . 5  |-  ( y  e.  { A ,  B }  ->  ( y  =  A  \/  y  =  B ) )
2120adantl 277 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  (
y  =  A  \/  y  =  B )
)
2213, 19, 21mpjaodan 799 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  -.  B  <  y )
232, 3, 5, 22supmaxti 7016 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B )
2423ex 115 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1363    e. wcel 2158   {cpr 3605   class class class wbr 4015   supcsup 6994   RR*cxr 8004    < clt 8005    <_ cle 8006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-pre-ltirr 7936  ax-pre-apti 7939
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-iota 5190  df-riota 5844  df-sup 6996  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011
This theorem is referenced by:  xrmaxltsup  11279  xrmaxadd  11282  xrmineqinf  11290
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